# Neural Network Approximations for Calabi-Yau Metrics

arXiv: https://arxiv.org/abs/2012.15821 · HTML: https://mlsanity.com/paper/2012.15821/

## Key facts

- Cited by 32 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 31 Dec 2020.

## Built on by (most cited first, 30 of 32)

- [Learning Size and Shape of Calabi-Yau Spaces](https://mlsanity.com/paper/2111.01436/) (2021)
- [Machine-Learning Mathematical Structures](https://mlsanity.com/paper/2101.06317/) (2021)
- [Machine Learning Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2112.06350/) (2021)
- [Precision Machine Learning](https://mlsanity.com/paper/2210.13447/) (2022)
- [Machine Learned Calabi-Yau Metrics and Curvature](https://mlsanity.com/paper/2211.09801/) (2022)
- [Rigor with Machine Learning from Field Theory to the Poincar'e Conjecture](https://mlsanity.com/paper/2402.13321/) (2024)
- [Calabi-Yau Metrics, Energy Functionals and Machine-Learning](https://mlsanity.com/paper/2112.10872/) (2021)
- [Cluster Algebras: Network Science and Machine Learning](https://mlsanity.com/paper/2203.13847/) (2022)
- [Metric Flows with Neural Networks](https://mlsanity.com/paper/2310.19870/) (2023)
- [Machine Learning Kreuzer--Skarke Calabi--Yau Threefolds](https://mlsanity.com/paper/2112.09117/) (2021)
- [Physics-informed neural network solves minimal surfaces in curved spacetime](https://mlsanity.com/paper/2509.10866/) (2025)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [A Machine Learning Approach to the Nirenberg Problem](https://mlsanity.com/paper/2602.12368/) (2026)
- [Learning Discretized Neural Networks under Ricci Flow](https://mlsanity.com/paper/2302.03390/) (2023)
- [TASI Lectures on Physics for Machine Learning](https://mlsanity.com/paper/2408.00082/) (2024)
- [Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm](https://mlsanity.com/paper/2410.11284/) (2024)
- [Symbolic Approximations to Ricci-flat Metrics Via Extrinsic Symmetries of Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2412.19778/) (2024)
- [Machine Learning Algebraic Geometry for Physics](https://mlsanity.com/paper/2204.10334/) (2022)
- [cymyc -- Calabi-Yau Metrics, Yukawas, and Curvature](https://mlsanity.com/paper/2410.19728/) (2024)
- [Thoughts on the Consistency between Ricci Flow and Neural Network Behavior](https://mlsanity.com/paper/2111.08410/) (2021)
- [From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook](https://mlsanity.com/paper/2202.06086/) (2022)
- [Minimal surfaces, Knots, and Neural Networks](https://mlsanity.com/paper/2605.26234/) (2026)
- [Intelligent Explorations of the String Theory Landscape](https://mlsanity.com/paper/2204.08073/) (2022)
- [Machine Learning on generalized Complete Intersection Calabi-Yau Manifolds](https://mlsanity.com/paper/2209.10157/) (2022)
- [Hamiltonian Neural Networks approach to fuzzball geodesics](https://mlsanity.com/paper/2502.20881/) (2025)
- [Approximate Ricci-flat Metrics for Calabi-Yau Manifolds](https://mlsanity.com/paper/2506.15766/) (2025)
- [Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation](https://mlsanity.com/paper/2602.07834/) (2026)
- [Black holes and the loss landscape in machine learning](https://mlsanity.com/paper/2306.14817/) (2023)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)
- [Learning to be Simple](https://mlsanity.com/paper/2312.05299/) (2023)

## Abstract (arXiv)

Ricci flat metrics for Calabi-Yau threefolds are not known analytically. In this work, we employ techniques from machine learning to deduce numerical flat metrics for the Fermat quintic, for the Dwork quintic, and for the Tian-Yau manifold. This investigation employs a single neural network architecture that is capable of approximating Ricci flat Kaehler metrics for several Calabi-Yau manifolds of dimensions two and three. We show that measures that assess the Ricci flatness of the geometry decrease after training by three orders of magnitude. This is corroborated on the validation set, where the improvement is more modest. Finally, we demonstrate that discrete symmetries of manifolds can be learned in the process of learning the metric.